The Blues have slumped to sixth in the Premier League table after suffering a 6-0 loss at the Etihad Stadium – their worst defeat since April 1991.
Sergio Aguero hit a hat-trick, Raheem Sterling grabbed a brace and Ilkay Gundogan was on target as Chelsea collapsed.
Hazard, who is stalling on a contract extension, is being targeted by Real Madrid.
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The Champions League winners fancy the Belgian as a replacement for Cristiano Ronaldo, who departed last summer.
Chelsea head coach Maurizio Sarri has urged Hazard to make a decision on his future.
And Blues supporters reckon he would be better off by departing.
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Writing on Twitter, one fan said: “Hi @hazardeden10 please leave. You deserve a better club.”
A second wrote: “Leave, you don’t deserve this @hazardeden10.”
Another fan commented: “Kante & Hazard leave. You deserve better.”
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While a fourth moaned: “Quality players like hazard, Cho, Rudiger, Kante and Higuain must leave this average club. Don’t waste ur talent here.”
Hazard revealed last week that he has already made up his mind over his future.
“I know what I am going to do,” he told French radio station RMC. “I have made a decision.”